Maximum Likelihood

Maximum likelihood, also called the maximum likelihood method, is the procedure of finding the value of one or more parameters for a given statistic which makes the known likelihood distribution a maximum. The maximum likelihood estimate for a parameter is denoted .

For a Bernoulli distribution,

(1)

so maximum likelihood occurs for . If is not known ahead of time, the likelihood function is

			(2)
			(3)
			(4)

where or 1, and , ..., .

(5)

(6)

Rearranging gives

(7)

(8)

For a normal distribution,

			(9)
			(10)

(11)

and

(12)

giving

(13)

Similarly,

(14)

gives

(15)

Note that in this case, the maximum likelihood standard deviation is the sample standard deviation, which is a biased estimator for the population standard deviation.

For a weighted normal distribution,

(16)

(17)

(18)

gives

mu^^=(sum(x_i)/(sigma_i^2))/(sum1/(sigma_i^2)).

(19)

The variance of the mean is then

(20)

But

(21)

			(22)
			(23)
			(24)

For a Poisson distribution,

(25)

(26)

(27)

(28)

References

Harris, J. W. and Stocker, H. "Maximum Likelihood Method." §21.10.4 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 824, 1998.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Least Squares as a Maximum Likelihood Estimator." §15.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 651-655, 1992.

Maximum Likelihood

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